M38 Lec 111213

MATH 38Mathematical Analysis III

I. F. Evidente

IMSP (UPLB)

November 12, 2013

Outline

1 Sets of Numbers

2 Laws on Exponents

3 Review of some limitsRational FunctionsSome transcendental functionsComposition of functionsHandling indeterminate formsSqueeze Theorem

Leonhard Euler (1707-1783)

Some very important numbers

e, i ,pi,1,0

Which are counting numbers?Which are integers?Which are rationals?Which are irrational?Which are real numbers?Which are imaginary?Which are complex?


e, i ,pi,1,0

Which are counting numbers?

Which are integers?Which are rationals?Which are irrational?Which are real numbers?Which are imaginary?Which are complex?


e, i ,pi,1,0

Which are counting numbers?Which are integers?

Which are rationals?Which are irrational?Which are real numbers?Which are imaginary?Which are complex?


e, i ,pi,1,0

Which are counting numbers?Which are integers?Which are rationals?

Which are irrational?Which are real numbers?Which are imaginary?Which are complex?


e, i ,pi,1,0

Which are counting numbers?Which are integers?Which are rationals?Which are irrational?

Which are real numbers?Which are imaginary?Which are complex?


e, i ,pi,1,0

Which are counting numbers?Which are integers?Which are rationals?Which are irrational?Which are real numbers?

Which are imaginary?Which are complex?


e, i ,pi,1,0

Which are counting numbers?Which are integers?Which are rationals?Which are irrational?Which are real numbers?Which are imaginary?

Which are complex?


e, i ,pi,1,0

Which are counting numbers?Which are integers?Which are rationals?Which are irrational?Which are real numbers?Which are imaginary?Which are complex?

If time allows, we will prove this

e i x = cosx+ i sinx

Sets of Numbers

counting numbers:

N= {1,2,3, ...}

integers:Z= {...,2,1,0,1,2, ...}

rationals:Q=

{p

q| p,q Z and q 6= 0

}

irrationals:e,pi,

p2 Q

reals:R=QQ

Sets of Numbers

counting numbers:N= {1,2,3, ...}

integers:Z= {...,2,1,0,1,2, ...}

rationals:Q=

{p

q| p,q Z and q 6= 0

}

irrationals:e,pi,

p2 Q

reals:R=QQ

Sets of Numbers


integers:

Z= {...,2,1,0,1,2, ...}

rationals:Q=

{p

q| p,q Z and q 6= 0

}

irrationals:e,pi,

p2 Q

reals:R=QQ

Sets of Numbers


integers:Z= {...,2,1,0,1,2, ...}

rationals:Q=

{p

q| p,q Z and q 6= 0

}

irrationals:e,pi,

p2 Q

reals:R=QQ

Sets of Numbers


integers:Z= {...,2,1,0,1,2, ...}

rationals:

Q={p

q| p,q Z and q 6= 0

}

irrationals:e,pi,

p2 Q

reals:R=QQ

Sets of Numbers


integers:Z= {...,2,1,0,1,2, ...}

rationals:Q=

{p

q| p,q Z and q 6= 0

}

irrationals:e,pi,

p2 Q

reals:R=QQ

Sets of Numbers


integers:Z= {...,2,1,0,1,2, ...}

rationals:Q=

{p

q| p,q Z and q 6= 0

}

irrationals:

e,pi,p2 Q

reals:R=QQ

Sets of Numbers


integers:Z= {...,2,1,0,1,2, ...}

rationals:Q=

{p

q| p,q Z and q 6= 0

}

irrationals:e,pi,

p2 Q

reals:R=QQ

Sets of Numbers


integers:Z= {...,2,1,0,1,2, ...}

rationals:Q=

{p

q| p,q Z and q 6= 0

}

irrationals:e,pi,

p2 Q

reals:

R=QQ

Sets of Numbers


integers:Z= {...,2,1,0,1,2, ...}

rationals:Q=

{p

q| p,q Z and q 6= 0

}

irrationals:e,pi,

p2 Q

reals:R=QQ

Outline

1 Sets of Numbers

2 Laws on Exponents


Laws on Exponents

1 anam =

an+m

2an

am= anm

3 (ab)n = anbn4

(ab

)n= a

n

bn5 (an)m = anm

Laws on Exponents

1 anam = an+m

2an

am= anm

3 (ab)n = anbn4

(ab

)n= a

n

bn5 (an)m = anm

Laws on Exponents

1 anam = an+m2

an

am=

anm

3 (ab)n = anbn4

(ab

)n= a

n

bn5 (an)m = anm

Laws on Exponents

1 anam = an+m2

an

am= anm

3 (ab)n = anbn4

(ab

)n= a

n

bn5 (an)m = anm

Laws on Exponents

1 anam = an+m2

an

am= anm

3 (ab)n =

anbn

4

(ab

)n= a

n

bn5 (an)m = anm

Laws on Exponents

1 anam = an+m2

an

am= anm

3 (ab)n = anbn

4

(ab

)n= a

n

bn5 (an)m = anm

Laws on Exponents

1 anam = an+m2

an

am= anm

3 (ab)n = anbn4

(ab

)n=

an

bn5 (an)m = anm

Laws on Exponents

1 anam = an+m2

an

am= anm

3 (ab)n = anbn4

(ab

)n= a

n

bn

5 (an)m = anm

Laws on Exponents

1 anam = an+m2

an

am= anm

3 (ab)n = anbn4

(ab

)n= a

n

bn5 (an)m =

anm

Laws on Exponents

1 anam = an+m2

an

am= anm

3 (ab)n = anbn4

(ab

)n= a

n

bn5 (an)m = anm

Let n N(1)n =

{

1 if n is even1 if n is odd

1n ={1 if n is even1 if n is odd

Let n N(1)n =

{1 if n is even

1 if n is odd


Let n N(1)n =

{1 if n is even1 if n is odd


Let n N(1)n =


1n ={

1 if n is even1 if n is odd

Let n N(1)n =


1n ={1 if n is even

1 if n is odd

Let n N(1)n =



Examples

1 (4)(2x)n1

2

(1

5

)n (4x5

)n13

(13

)n (2x3

)n

Outline

1 Sets of Numbers

2 Laws on Exponents


Limits of Rational Functions

Rational Function

h(x)= f (x)g (x)

where f and g are polynomial functions and g is not the constant zerofunction.


Evaluate the following limits:

1 limx

2x3+4x2+26x35x =

1

3

2 limx

2x5+4x2+26x35x =

3 limx

2x3+4x2+26x55x = 0



1 limx

2x3+4x2+26x35x =

1

3

2 limx

2x5+4x2+26x35x =

3 limx

2x3+4x2+26x55x =

0



1 limx

2x3+4x2+26x35x =

1

3

2 limx

2x5+4x2+26x35x =

3 limx

2x3+4x2+26x55x = 0

Limits of Transcendental Functions

1 limxe

x =

2 limx2

x =

0

3 limx lnx =

4 limx0+

lnx =

5 limxArctanx =

pi

2

6 limxArctanx =

pi2


1 limxe

x =2 lim

x2x =

0

3 limx lnx =

4 limx0+

lnx =

5 limxArctanx =

pi

2

6 limxArctanx =

pi2


1 limxe

x =2 lim

x2x = 0

3 limx lnx =

4 limx0+

lnx =

5 limxArctanx =

pi

2

6 limxArctanx =

pi2


1 limxe

x =2 lim

x2x = 0

3 limx lnx =

4 limx0+

lnx =

5 limxArctanx =

pi

2

6 limxArctanx =

pi

2

Recall a theorem...

TheoremIf lim

xa g (x)= L and f is continuous at L, then

limxa f (g (x))

= f (L)

Recall a theorem...

TheoremIf lim

xa g (x)= L and f is continuous at L, then

limxa f (g (x))= f (L)

Examples

1 limxcos

(1

x

)=

1

2 limx ln

( xx+1

)= 0

Examples

1 limxcos

(1

x

)= 1

2 limx ln

( xx+1

)= 0

Examples

1 limxcos

(1

x

)= 1

2 limx ln

( xx+1

)=

0

Examples

1 limxcos

(1

x

)= 1

2 limx ln

( xx+1

)= 0

As a consequence...

CorollaryIf lim

xa g (x)= L and f is continuous everywhere, then

limxa f (g (x))= L

Can you give some examples of functions that are continuous everywhere?

Examples

1 limxe

1/x2 =

1

2 limxsin

(2

ex

)= 0

3 limxe

x = 0

Examples

1 limxe

1/x2 = 1

2 limxsin

(2

ex

)= 0

3 limxe

x = 0

Examples

1 limxe

1/x2 = 1

2 limxsin

(2

ex

)=

0

3 limxe

x = 0

Examples

1 limxe

1/x2 = 1

2 limxsin

(2

ex

)= 0

3 limxe

x = 0

Examples

1 limxe

1/x2 = 1

2 limxsin

(2

ex

)= 0

3 limxe

x =

0

Examples

1 limxe

1/x2 = 1

2 limxsin

(2

ex

)= 0

3 limxe

x = 0

Also...

Forms:(e):

limit is (e): limit is 0(ln()): limit is (ln(0+)): limit is (Arctan()): limit is pi2(Arctan()): limit is pi2

Also...

Forms:(e): limit is

(e): limit is 0(ln()): limit is (ln(0+)): limit is (Arctan()): limit is pi2(Arctan()): limit is pi2

Also...

Forms:(e): limit is (e):

limit is 0(ln()): limit is (ln(0+)): limit is (Arctan()): limit is pi2(Arctan()): limit is pi2

Also...

Forms:(e): limit is (e): limit is 0

(ln()): limit is (ln(0+)): limit is (Arctan()): limit is pi2(Arctan()): limit is pi2

Also...

Forms:(e): limit is (e): limit is 0(ln()):

limit is (ln(0+)): limit is (Arctan()): limit is pi2(Arctan()): limit is pi2

Also...

Forms:(e): limit is (e): limit is 0(ln()): limit is

(ln(0+)): limit is (Arctan()): limit is pi2(Arctan()): limit is pi2

Also...

Forms:(e): limit is (e): limit is 0(ln()): limit is (ln(0+)):

limit is (Arctan()): limit is pi2(Arctan()): limit is pi2

Also...

Forms:(e): limit is (e): limit is 0(ln()): limit is (ln(0+)): limit is

(Arctan()): limit is pi2(Arctan()): limit is pi2

Also...

Forms:(e): limit is (e): limit is 0(ln()): limit is (ln(0+)): limit is (Arctan()):

limit is pi2(Arctan()): limit is pi2

Also...

Forms:(e): limit is (e): limit is 0(ln()): limit is (ln(0+)): limit is (Arctan()): limit is pi2

(Arctan()): limit is pi2

Also...

Forms:(e): limit is (e): limit is 0(ln()): limit is (ln(0+)): limit is (Arctan()): limit is pi2(Arctan()):

limit is pi2

Also...

Forms:(e): limit is (e): limit is 0(ln()): limit is (ln(0+)): limit is (Arctan()): limit is pi2(Arctan()): limit is pi2

Examples

1 limx ln

(1

x

)=

2 limx0+

Arctan(lnx)= pi2

Examples

1 limx ln

(1

x

)=

2 limx0+

Arctan(lnx)=

pi2

Examples

1 limx ln

(1

x

)=

2 limx0+

Arctan(lnx)= pi2

Some indeterminate forms

1 limxx sin

(pix

)=

pi

2 limx

(x+1x

)x= e


1 limxx sin

(pix

)=pi

2 limx

(x+1x

)x= e


1 limxx sin

(pix

)=pi

2 limx

(x+1x

)x=

e


1 limxx sin

(pix

)=pi

2 limx

(x+1x

)x= e

Squeeze Theorem

TheoremSuppose the functions f , g and h are defined on some open interval Icontaining a, except possibly at a itself, and that

f (x) g (x) h(x) for all x in I for which x 6= alimxa f (x)= limxa g (x)= L.

Then limxa g (x)= L.

Squeeze Theorem

1 limxsinx =

dne

2 limx

sinx

x= 0

3 limx

cosx

x= 0

4 limx

3x+cosx34x =

3

4

Squeeze Theorem

1 limxsinx = dne

2 limx

sinx

x= 0

3 limx

cosx

x= 0

4 limx

3x+cosx34x =

3

4

Squeeze Theorem

1 limxsinx = dne

2 limx

sinx

x=

0

3 limx

cosx

x= 0

4 limx

3x+cosx34x =

3

4

Squeeze Theorem

1 limxsinx = dne

2 limx

sinx

x= 0

3 limx

cosx

x= 0

4 limx

3x+cosx34x =

3

4

Squeeze Theorem

1 limxsinx = dne

2 limx

sinx

x= 0

3 limx

cosx

x=

0

4 limx

3x+cosx34x =

3

4

Squeeze Theorem

1 limxsinx = dne

2 limx

sinx

x= 0

3 limx

cosx

x= 0

4 limx

3x+cosx34x =

3

4

Squeeze Theorem

1 limxsinx = dne

2 limx

sinx

x= 0

3 limx

cosx

x= 0

4 limx

3x+cosx34x =

34

Squeeze Theorem

1 limxsinx = dne

2 limx

sinx

x= 0

3 limx

cosx

x= 0

4 limx

3x+cosx34x =

3

4

Sets of NumbersLaws on ExponentsReview of some limitsRational FunctionsSome transcendental functionsComposition of functionsHandling indeterminate formsSqueeze Theorem

M38 Lec 111213

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Transcript of M38 Lec 111213